1. **State the problem:** Simplify the expression $3x \times \frac{1}{3} y$. 2. **Recall the multiplication rule:** When multiplying terms, multiply the coefficients (numbers) and then multiply the variables. 3. **Multiply the coefficients:** $$3 \times \frac{1}{3} = \frac{3}{1} \times \frac{1}{3} = \frac{\cancel{3}}{1} \times \frac{1}{\cancel{3}} = 1$$ 4. **Multiply the variables:** $$x \times y = xy$$ 5. **Combine the results:** $$1 \times xy = xy$$ **Final answer:** $xy$